Incomplete lecture notes for the course of
QUANTUM MECHANICS 20015/2016
M. Damnjanovi´c
Belgrade, 2012
PREFACE
Tekst je i ove godine dopunjavan delovima predavanja, delimicno reorganizovan. Jedan
deo zadataka sa veˇ
zbi je promenjen, pre svega zbog promene asistenta na predmetu (Marko
Milivojevi´
c), ali i zbog naˇ
cina selekcije. I dalje se ovo moˇ
ze smatrati samo prate´
cim
materijalom predavanja, a nikako ne kompletnim i proˇ
ciˇs´
cenim odrazom ispredavanih
lekcija. To ´
ce, nadamo se, postati slede´
cih godina. Tekst dobrim delom nije proveravan
nakon pisanja, a posebno ne lektorisan. Stoga treba biti skeptiˇ
can prema formulama (!),
pa ˇ
cak i nekim formulacijama, jer prilikom elektronskog editovanja svaka nepaˇ
znja moˇ
ze
da bude vrlo kreativna. Dalje, nekompletan je: mada su sva predavanja obradjena, i
u tom smislu tekst pokriva kurs, ˇ
ceste indikacije podnaslova ukazuju na projektovana
proˇsirenja neophodna za potpunije sagledavanje pojedinih tema, otkrivaju´
ci najvaˇ
znije
konceptualne celine koje kursom nisu obuhva´
cene. Slike i primeri koji treba da ilustruju
sadrˇ
zaje krajnje su redukovani. Zato molim studente da tekst bude pre svega podsetnik
za ono sto treba uraditi, a da za konaˇ
cnu pripremu ispita koriste dopunsku literaturu.
3.10.2015, M.D.
i
CONTENTS
iii
2.7
Feynman’s path integral approach
. . . . . . . . . . . . . . . . . . . . . . .
34
3
Galilean Transformations
36
3.1
Galilean Group
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.2
Galilean transformations of the classical variables
. . . . . . . . . . . . . .
37
3.3
Quantization: Wigner theorem
. . . . . . . . . . . . . . . . . . . . . . . . .
39
3.4
Quantization of the Galilean group
. . . . . . . . . . . . . . . . . . . . . .
39
3.5
Active and passive interpretations
. . . . . . . . . . . . . . . . . . . . . . .
41
4
Rotations and Angular Momentum
43
4.1
Elementary Properties of Rotations
. . . . . . . . . . . . . . . . . . . . . .
43
4.2
Algebra of angular momentum
. . . . . . . . . . . . . . . . . . . . . . . . .
44
4.2.1
Irreducible representations of angular momentum
. . . . . . . . . .
45
4.2.2
Square of the angular momentum
. . . . . . . . . . . . . . . . . . .
47
4.2.3
Discussion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.3
Orbital Angular Momentum
. . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.3.1
Coordinate representation
. . . . . . . . . . . . . . . . . . . . . . .
49
4.3.2
Standard Basis: Spherical Harmonics
. . . . . . . . . . . . . . . . .
50
4.4
Central potentials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
4.4.1
Free particle
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
4.4.2
Coulomb potential and Hydrogen like atoms
. . . . . . . . . . . . .
53
4.5
Spin
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
4.5.1
Zeeman’s Effect
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
4.5.2
Interior degrees of freedom
. . . . . . . . . . . . . . . . . . . . . . .
55
4.5.3
Formalism of the spin
s
=
1
2
. . . . . . . . . . . . . . . . . . . . . .
56
4.6
Addition of angular momenta
. . . . . . . . . . . . . . . . . . . . . . . . .
57
4.6.1
Irreducible subspaces and standard basis
. . . . . . . . . . . . . . .
57
4.6.2
Examples and applications
. . . . . . . . . . . . . . . . . . . . . . .
62
4.6.3
L
−
S
and
j
−
j
coupling
. . . . . . . . . . . . . . . . . . . . . . . .
64
5
Identical Particles
68
5.1
Quantum Formalism
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
5.1.1
Permutational Indistinguishability and Symmetrization
. . . . . . .
68
5.1.2
Structure of the state space — occupation numbers
. . . . . . . . .
70
5.1.3
States
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
5.1.4
Operators
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5.2
Second Quantization
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.2.1
Fock space
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.2.2
Creation and annihilation operators
. . . . . . . . . . . . . . . . . .
81
5.2.3
Bosonic and fermionic commutation relations
. . . . . . . . . . . .
82
5.2.4
Representation of the second quantization
. . . . . . . . . . . . . .
83
6
Approximate Methods
85
6.1
Time Independent Perturbations
. . . . . . . . . . . . . . . . . . . . . . .
85
6.1.1
Perturbed and Unperturbed Hamiltonian
. . . . . . . . . . . . . . .
85
6.1.2
Perturbative expansion
. . . . . . . . . . . . . . . . . . . . . . . . .
86
iv
CONTENTS
6.1.3
Higher Corrections for Non-degenerate Level
. . . . . . . . . . . . .
87
6.2
Adiabatic approximation
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
6.3
Variational Method
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
6.3.1
General characteristics
. . . . . . . . . . . . . . . . . . . . . . . . .
89
6.3.2
Hartree-Fock method
. . . . . . . . . . . . . . . . . . . . . . . . . .
92
6.4
Density Functional Theory
. . . . . . . . . . . . . . . . . . . . . . . . . . .
95
6.4.1
Uniform electron gas
. . . . . . . . . . . . . . . . . . . . . . . . . .
98
6.4.2
Exchange energy
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
6.5
Time dependent perturbations
. . . . . . . . . . . . . . . . . . . . . . . . .
99
6.5.1
Expansion of the evolution
. . . . . . . . . . . . . . . . . . . . . . . 100
6.5.2
Transition amplitudes
. . . . . . . . . . . . . . . . . . . . . . . . . 101
6.5.3
Transition probabilities
. . . . . . . . . . . . . . . . . . . . . . . . . 102
6.6
Elementary scattering theory
. . . . . . . . . . . . . . . . . . . . . . . . . 105
A Technical support
107
A.1 Separation of Variables
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.2 Hypergeometric equation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
B Solutions of Exercises
110
B.0.1
Quantum Kinematics
. . . . . . . . . . . . . . . . . . . . . . . . . . 110
B.0.2
Quantum dynamcs
. . . . . . . . . . . . . . . . . . . . . . . . . . . 112
B.0.3
Galilean transformations
. . . . . . . . . . . . . . . . . . . . . . . . 113
B.0.4
Angular momentum
. . . . . . . . . . . . . . . . . . . . . . . . . . 114
B.0.5
Identical particles
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2
CHAPTER 1. QUANTUM KINEMATICS
properties, as well as the primary notion of the system. Consequently, two ensembles are
in the same state if all the measurements performed on them give the same results. Again,
the notion of the state is dependent on our knowledge of the possible properties of the
system. Important example is that before discovery of the spin, ensembles (i.e. states)
differing only in the value of the spin were considered the same.
Intuitively,
measurement
is any process of determining of some property of an ensem-
ble. In fact, the very idea of a property of the system is that it is observable, i.e. measur-
able. This means that for each property (observable)
A
there is at least one measuring
device (in this sense, the notions of the physical quantity and of the device measuring
it may be identified), apparatus
A
, capable to distinguish between various values of
A
:
each value corresponds to a particular position of the apparatus’ pointer. In other words,
in the course of the interaction of the apparatus with the measured system some of these
values is realized. Two important facts should be emphasized in this context. Firstly,
various members of the same ensemble may produce different pointer positions. There-
fore, measurements on the ensemble necessarily have statistical nature. Secondly, before
measurement is performed nothing can be said about the measured property. This may
be interpreted such that the measured property does not exist without the apparatus, or
even that the property is realized or imposed by measurement.
The statistical nature of the measurement is well known even within the classical
framework: a measurement does not give the result with certainty, but many measure-
ments are performed, and the result is obtained by statistical analysis. Therefore, a
single measurement is meaningless. Further, since the measurement is an interaction of
apparatus and system, it may change, or even destruct the system (this is particularly
important for small systems), and it may be impossible to repeat the procedure on the
same system. Therefore, in general, measurements are performed on the ensembles of
systems. Nevertheless, as it will be stressed out in the analysis of double slit experiment
(Subsection
1.1.2
), the necessity for the statistical approach and ensembles in quantum
mechanics stems from additional, quite substantial reason.
To summarize, measurement means
measurement of some physical observable
A
on
the ensemble (in the state)
ρ
, giving as the result probability distribution of the possible
values of
A
. Precisely, let
σ
(
A
) =
{
a
1
, a
2
, . . .
}
be the set of the possible values of
A
(i.e. of
the positions of the pointer of the apparatus, defined independently of
ρ
); each particular
system from the measured ensemble
ρ
interacts with apparatus successively, and due to
this interaction the pointer gets a series of values
a
i
from
σ
(
A
). Let in the course of this
measurement each value
a
i
is pointed to altogether
N
i
times. Obviously,
N
=
∑
i
N
i
is the
number of systems in the ensemble. Then the result of this measurement is the probability
distribution
v
(
a
i
, A, ρ
)
def
=
N
i
/N
. Since the only criterion of the validity of any physical
theory (and particularly quantum mechanics) is the comparison with the experiment, the
fundamental task of such a theory is to give prediction for
v
(
a
i
, A, ρ
) for each
ρ
and
A
in
terms of its formalism. Before proceeding further in this direction, several remarks should
be made.
Firstly, we note here that the (quantum) theory can be well founded only with infinite
ensembles, and in the rest of the text this will be always assumed. However, in the real
experiments
N
must be finite, but large enough to provide reliable statistics, i.e. the
statistics enabling comparison to the theoretical
N
=
∞
limit.
